Defining polynomial
|
\(x^{23} + 940\)
|
Invariants
| Base field: | $\Q_{47}$ |
| Degree $d$: | $23$ |
| Ramification exponent $e$: | $23$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{47}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 47 }) }$: | $23$ |
| This field is Galois and abelian over $\Q_{47}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 47 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{47}$ |
| Relative Eisenstein polynomial: |
\( x^{23} + 940 \)
|
Ramification polygon
| Residual polynomials: | $z^{22} + 23z^{21} + 18z^{20} + 32z^{19} + 19z^{18} + 44z^{17} + 38z^{16} + 5z^{15} + 10z^{14} + z^{13} + 39z^{12} + 29z^{11} + 29z^{10} + 39z^{9} + z^{8} + 10z^{7} + 5z^{6} + 38z^{5} + 44z^{4} + 19z^{3} + 32z^{2} + 18z + 23$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{23}$ (as 23T1) |
| Inertia group: | $C_{23}$ (as 23T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $23$ |
| Wild slopes: | None |
| Galois mean slope: | $22/23$ |
| Galois splitting model: | Not computed |